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Published in Vadose Zone Journal 3:947-955 (2004)
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA

ORIGINAL RESEARCH

Joint Distributions of the Unsaturated Soil Hydraulic Parameters and their Effect on Other Variates

Gerrit H. de Rooija,*, Roy T. A. Kasteelb, Andreas Papritzc and Hannes Flühlerc

a Wageningen Univ., Dep. Environ. Sci., Sub-Dep. Water Resources, Soil Physics, Agrohydrology, and Groundwater Management Group, Nieuwe Kanaal 11, 6709 PA The Netherlands
b Forschungszentrum Jülich GmbH, Institute of Chemistry and Dynamics of the Geosphere IV, Agrosphere, 52425 Jülich, Germany
c Institute of Terrestrial Ecology, Soil Physics, Federal Institute of Technology Zürich, Grabenstraße 3, 8952 Schlieren, Switzerland
* Corresponding author (ger.derooij{at}wur.nl)

Received 24 December 2003.



   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSIONS
 REFERENCES
 
Variability in the unsaturated soil hydraulic properties affects the behavior of water and solutes in the subsurface. When these properties are described by parametric functions, subsurface heterogeneity can be quantified by variations in the parameters of these functions. We propose a procedure to characterize systematically the statistical properties of each soil hydraulic parameter and the parameter correlations involved. First, the parameter with the best-defined probability distribution function (pdf) is identified. Curvilinear regression analysis identifies transformations that linearize the relationships between this reference parameter and the remaining parameters. From this we determine the joint pdf of the transformed parameters, which is then used to calculate the distributions of derived variates (functions of the hydraulic parameters). The procedure is applied to the hydraulic parameters of 140 samples in total taken from two layers of an agricultural soil profile. We developed analytical expressions for the pdfs of the derived variates (hydraulic conductivities and water contents at pressure heads between 0 and –5000 cm) from the multivariate parameter pdf. The numerical integration required to evaluate these expressions proved extremely cumbersome, thus reducing the robustness of the analytical expressions. We therefore performed a Monte Carlo simulation from which we determined the first two moments, as well as the skewness and excess of the derived variates. The estimated mean and standard deviation of the derived variates generally agreed well with values determined directly from the soil samples. Skewness and excess were estimated less accurately. Both the derived analytical pdfs and the Monte Carlo simulation showed markedly nonsymmetrical pdfs, suggesting that it is generally insufficient to limit statistical analyses to the first two moments only.

Abbreviations: pdf, probability distribution function • SHP, soil hydraulic parameter • vG parameters, SHPs according to van Genuchten (1980)


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSIONS
 REFERENCES
 
SPATIAL VARIATION in the soil water retention and unsaturated hydraulic conductivity properties affects unsaturated water flow and solute transport (Nielsen et al., 1973; Biggar and Nielsen, 1976). Consequently, researchers often adopt a stochastic approach to model field-scale water flow and solute transport (e.g., Russo et al., 1998, and the reviews by Russo, 1997, and Yeh, 1998). The soil water retention and hydraulic conductivity curves are usually described by means of parametric functions. Soil variation can then be described by variation of the parameters of these functions.

Several studies addressed different aspects of the joint variation of soil hydraulic parameters (SHPs). Russo and Bresler (1982) modeled soil heterogeneity by a joint distribution of three hydraulic parameters. A field was approximated by a set of uniform soil columns in which water flow and solute transport were described by highly simplified one-dimensional models. Each column was assigned three random soil hydraulic parameters. Hills et al. (1992) focused on the tradeoff between the computational benefit of keeping some parameters constant and the loss of accuracy when predicting the degree of saturation at various pressure heads. Some scatter plots indicating correlations between parameters were presented but without the systematic framework that will be developed in this paper.

Carsel and Parrish (1988) employed the Johnson system of transformations to transform the empirical distributions of the SHPs of van Genuchten (1980) (vG parameters) to Gaussianity. Mallants et al. (1996) applied seven transformations, including some proposed by Carsel and Parrish (1988), to vG parameters measured on 180 samples taken from a 31-m transect. For each parameter, they retained the transformation that resulted in the best approximation of a Gaussian distribution. The correlation and covariance between the transformed parameters were determined after the transformations had been defined.

The procedures of Carsel and Parrish (1988) and Mallants et al. (1996) identify parameter transformations for which appropriate theoretical distribution functions can be determined. They do so for each parameter independently, and quantify relationships between parameters a posteriori. Typically, several predefined transformations are used indiscriminately, and the transformation that gives the best result for a particular parameter is selected. In contrast to previous studies, in this paper we introduce a procedure that first identifies relationships between parameters. Since the distribution of the saturated hydraulic conductivity (Ks, L T–1) of many soils is excellently described by a lognormal distribution, our procedure will utilize this feature to find more successful transformations of the remaining parameters.

Our first objective is to rigorously develop the new method to determine the joint pdf of SHPs of a field soil. We shall demonstrate the method using the parameterization of van Genuchten (1980) because of its widespread application, but the method can be applied to any other parameterization. A comprehensive treatment of the joint variation of SHPs and its effect on the variation of other soil properties of interest is still lacking. Therefore, our second objective is to demonstrate the effect of the joint variation of the SHPs on derived soil properties of interest. We shall use the method of auxiliary variables to find the pdf of several such derived variates, including water contents and hydraulic conductivities at given pressure heads.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSIONS
 REFERENCES
 
Soil
Undisturbed soil samples were taken from an arable field near Andelfingen in northern Switzerland (Kasteel, 1997; Forrer et al., 1999). The soil, a coarse-loamy eutric Cambisol (mesic Eutrochrept) (Soil Survey Staff, 1988), was annually plowed to a depth of 0.3 m. In each of four profiles, and at each depth of 0.13, 0.38, 0.63, and 0.88 m, seven soil cores (5.5-cm height, 5.2-cm diam.) were taken, and three cores were collected at depths of 0.25, 0.50, and 0.75 m, respectively. The given depths refer to the centers of the samples. In the laboratory the soil cores were cast in a resin to avoid bypass flow along the sample wall. Eliminating a few samples that were damaged during handling left a total of 140 samples.

For 122 samples, the soil hydraulic conductivity was measured at a minimum of two pressure heads (mostly two to four) slightly below saturation. A modified version of the spray apparatus of Dirksen and Matula (1994) was used to apply water at the low rates required. Two tensiometers and a time domain reflectometry probe, mounted in the hardened resin, measured the pressure heads h (cm) at 1.5 and 5.4 cm above the core bottom and the volumetric water content {theta} in the middle of the core. The hydraulic conductivity K (cm d–1) was calculated from the pressure head gradient and the applied flux density. The saturated hydraulic conductivity, considered here as a fitting parameter, was obtained by fitting the hydraulic conductivity curve of each individual sample to the unsaturated conductivity data [log(K) vs. h].

The desorption branch of the water characteristic was measured in a pressure membrane apparatus. The applied pressure head ranged from –1 to –690 cm (for a few samples from –1 to –15000 cm). Kasteel (1997) gave details of the measurements. The observations are summarized in Fig. 1 and 2 .



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  		Fig. 1. Soil water retention data for the plough layer and the subsoil.

 


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  		Fig. 2. Unsaturated hydraulic conductivity data for the plough layer and the subsoil.

 
The soil water characteristic and the soil hydraulic conductivity curve were parameterized according to van Genuchten (1980):

[1]
and

[2]
respectively, where {alpha} (cm–1) and {nu} are shape parameters. The subscripts "r" and "s" denote residual and saturated values, respectively.

Exploratory Statistical Analysis
We prepared box plots and histograms for all parameters for each sampling depth separately. The profile was separated into a plough layer (comprising the 0.13- and 0.25-m sampling depths) and the subsoil (all remaining depths). All samples within each layer were pooled, and the statistical analysis was performed for both layers separately (Table 1). Histograms and box plots were drawn again, but now for both layers.


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  		Table 1. Descriptive statistics of the fitted soil hydraulic parameters according to van Genuchten (1980).

 
Our parameter fitting procedure often produced {theta}r values of zero. Closer inspection revealed that {theta}r was always zero in the plough layer. In the subsoil, {theta}r was often zero while the distribution of the nonzero values was irregular, resulting in a complicated histogram. The soil profile at these larger depths generally does not dry out much under the climatic conditions of our field site. Consequently, the soil properties at the dry end (for which {theta}r is important) are of less interest than those in the wet range. Therefore, we neither attempted to fit a distribution for {theta}r, nor did we establish correlations between {theta}r and the other parameters. To obtain a global overview of the dependence among the parameters without imposing any limitations on the nature of the dependence, we determined Spearman's rank correlation coefficient for all possible parameter combinations.

Parameter Transformations and the Identification of the pdfs
The exploratory analysis revealed a variety of distributions for the five vG parameters (Fig. 3) . Several methods exist to transform distributions to a desired form (usually from nonnormal to normal). When the underlying pdf is unknown (as is the case here), these methods are often laborious or to some extent intuitive (for a discussion of various techniques see Kendall and Stuart, 1977, p. 175–185, 1979, p. 496–497; Kendall et al., 1983, p. 102–104).




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  		Fig. 3. Histograms of the fitted van Genuchten (1980) parameters. For the plough layer, {theta}r was fitted at zero for all samples.

 
We sought the pdfs of a set of variates that were dependent in various degrees. The pdf of one of these variates could be identified with confidence. We developed a novel procedure that took full advantage of the known pdf of this reference variate. The first step was to use curvilinear regression to fit any suitable function of the reference variate to observed data pairs of the reference variate, and one of the remaining variates. A function was suitable if it could be rearranged to produce a linear expression between a transformation of the variate and the Gaussian transformation of the reference variate. For the soil hydraulic properties studied here, ln(Ks) was convincingly Gaussian, and we chose Ks to be the reference variate. We therefore fitted linear and power functions of Ks, K–1s, ln(Ks), or [ln(Ks)]–1 that could be rearranged to give transformed variates that are linear functions of ln(Ks) (see Table 2 of de Rooij et al., 1999; note that the remarks for no. 8 and 9 of that table are erroneous). This step was repeated for all variates.


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  		Table 2. Spearman's rank correlation coefficients for both layers. The number of observations is given in parentheses. The hydraulic conductivity was not measured in all samples, which is reflected in the smaller number of observations for Ks.

 
After rearranging to linearity as outlined above, the absence of curved trends was checked visually. For each parameter, the transformation with the highest R2 value was selected. Since the transformations linearized the relationships between the Gaussian transformation of the reference variate and the other variates, the pdfs of all variates after transformation coalesced into Gaussianity (e.g., Papoulis, 1991, p. 88). The parameters of the individual pdfs could then be determined using standard methods.

The joint variation of the transformed vG parameters could be described with a multivariate normal pdf (Kendall and Stuart, 1977, p. 372–374):

[3]
where f(x) is the joint pdf, x is the vector of transformed variates, µ is the vector of mean values of x, b is the number of variates, and V is the covariance matrix of the transformed variates. A prime indicates a transposed vector. The elements vij of V are given by (Kendall and Stuart, 1977, p. 375):

[4]
where vij represents the correlation coefficient of transformed variates xi and xj, and vi and vj their standard deviations.

It is important to note that the vast majority of field sites for which observed SHP distributions have been reported exhibit a lognormally distributed Ks. Furthermore, Ks is generally considered to be the most variable SHP, regardless of the selected parameterization, and irrespective of whether Ks was measured directly or fitted to data for h < 0 (e.g., Dagan and Bresler, 1979). In view of this it is very likely that our method to identify pdfs for SHPs can be applied to many other soils, with multivariate Gaussian pdfs arising from the Gaussianity of ln(Ks) for most soils.

Derived Variates
The pdf of any derived parameter could be found analytically from the joint pdf of transformed vG parameters by the method of auxiliary variables (Papoulis, 1991, p. 147 and p. 182–183). Probability distribution functions of {theta}(h) and K(h) were determined for h = 0, –10, –50, –100, –1000, and –5000 cm. We also determined the pdf of the available water content ({theta}avail), defined as {theta}(hfc) – {theta}(hwp), where hfc is the pressure head at field capacity (–333 cm), and hwp is the wilting point (–15000 cm). Since {theta} does not depend on Ks and K is independent of {theta}s (see Eq. [1] and [2]), we needed to express the pdf of any derived variate in terms of the joint pdf of three original variates: the transformations of {alpha}, {nu}, and either Ks or {theta}s.

We denote the three jointly distributed original variates by a vector x = (x1, x2, x3). The vector y = (y1, y2, y3) denotes three derived variates that are functions gi(x) of the original variates:

[5]

If the functions gi(x) are single-valued for any combination of values of x1, x2, and x3, the pdf of y is given by (Papoulis, 1991, p. 183):

[6]
where J(x1, x2, x3) is the Jacobian of the transformation:

[7]

Since we seek the pdf of only one variate, we assign auxiliary variables to y2 and y3: y2 = x2, y3= x3 (Papoulis, 1991, p. 147). The Jacobian then reduces to {partial}g1/{partial}x1. We can find the pdf of y by expressing {partial}g1/{partial}x1 strictly in terms of x1, x2, and x3, and then integrate out the dependencies on the auxiliary variables x2 and x3:

[8]

Monte Carlo Simulation
The numerical integrations required to evaluate the analytical pdfs of the derived variates proved very cumbersome, and we therefore evaluated the pdfs through Monte Carlo simulations. One-hundred virtual soil samples were generated from the multivariate Gaussian pdf of the transformed SHPs according to Cressie (1993)(p. 200–202):

[9]
where q is the vector with a generated random value for each of the transformed parameters, µ is the vector of mean values of the transformed vG parameters, L is a lower triangular matrix resulting from the Cholesky decomposition of the covariance matrix, and {epsilon} is a vector of independent normal variates with zero mean and unit variance. Each generated vector q is a realization of the vector of variates.


   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
SUMMARY AND CONCLUSIONS
 REFERENCES
 
Statistical Properties of the Soil Hydraulic Parameters according to van Genuchten (1980)
Rank correlations between the parameters were found to be generally weak to moderate (Table 2), indicating that there is little redundancy in the vG parameterization, even though the vG parameters cannot be assumed to be independent. A notable exception is the very strong correlation between {alpha} and v for the plough layer. Surprisingly, this correlation vanishes entirely for the subsoil. Correlations are generally weaker for the subsoil than the plough layer, especially those involving {nu}.

Table 3 gives the regression relationships between lnKs and the transformed vG parameters (except {theta}r) that showed the strongest overall correlation with lnKs for both layers. Inspection of the scatter plots of the transformed parameters (not shown here) indicated that any nonlinearity was largely removed in all cases. In general, the relationships for the subsoil were weaker than those for the plough layer. The most extreme case was that of {nu}; several relationships between Ks and {nu} with considerable correlation coefficients could be found for the plough layer, whereas for the subsoil, none of the applied transformations yielded R2 values larger than 0.003.


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  		Table 3. Curvilinear regression relationships between Ks and {theta}s, {alpha}, and v, together with the coefficients of determination (R2).

 
Based on the transformations in Table 3, we fitted normal pdfs for lnKs (denoted k), ln{theta}s (w), ln{alpha} (a), and ln(ln{nu}) (n) (Table 4). We subsequently determined the correlation matrix of the transformed parameters (Table 5). The means and standard deviations in Table 4 and the correlation matrix in Table 5 fully characterize the multivariate normal pdf (Eq. [3] and [4]).


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  		Table 4. Descriptive statistics and Kolmogorov–Smirnov test statistic (KS) of four transformed van Genuchten (1980) parameters. A small value of the KS statistic indicates a good approximation of the empirical distribution by a normal distribution. The significance (the probability that the KS statistic exceeds the observed value if the distribution is indeed normal) incorporates the Lilliefors correction for the one-sample version of the test.{dagger}

 

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  		Table 5. Correlation matrices (lower triangular part) for both layers. R values are given.

 
In Table 5, the correlation between k and a is only moderately strong. Therefore, the strict similar media concept (Miller and Miller, 1956) is of limited value for this soil, since similar media scaling implies a perfect correlation between the logarithms of Ks and {alpha}.

Note that the transformations are such that the physical ranges of the untransformed parameters (<0,{infty}> for Ks and {alpha}, <0,1> for {theta}s, and <1,{infty}> for {nu}) yield the appropriate range (<–{infty},{infty}>) for their respective normally distributed transformations in all cases except for {theta}s. Among the various other choices, the selected parameter transformations indeed gave the best approximations of normal distributions, as indicated by a one-sample Kolmogorov–Smirnov test (with the mean and standard deviation of the normal test distribution estimated from the sample population, Table 4).

Carsel and Parrish (1988) recommended using the SB transformation for loams and sandy loams (the textural classes of the soil layers in this study) to obtain a normal pdf for most vG parameters. Mallants et al. (1996) found the SB transformation to be the most successful of the seven they tested to normalize the observed distributions of the vG parameters of a sandy loam. The SB transformation is defined as

[10]
where xi is the transformed ith variate, zi is the untransformed ith variate, and zi,min and zi,max are the smallest and largest observed value of zi.

We applied the SB transformation to all four vG parameters that we included in our analysis. A comparison of the correlation matrices showed that the correlation coefficients of SB-transformed parameters in most cases were less than those resulting from the transformations found by us. The results of a one-sample Kolmogorov–Smirnov test indicated approximations of normality that were generally poorer than those of our transformations, except for {alpha}. Inspection of the normal Q–Q plots (in which the quantiles of the observed distribution are plotted against those of the fitted distribution) revealed that the normal pdf fitted to the SB-transformed observations of {alpha} was likely to underestimate the occurrence of large values of {alpha}. Also, since the observed range is used to define the transformation (see Eq. [10]), generated values can never be outside that range. We therefore retained the fitted log-normal pdf for {alpha} in the remainder of the study.

To determine parameter correlations properly, it is crucial that the depth of the transitions between the soil layers be correctly identified. In an earlier study (de Rooij et al., 1999) we distinguished soil layers on the basis of the parameter statistics of all sampling depths. The depth of the plough layer was overestimated, resulting in a severe underestimation of virtually all parameter correlation coefficients of both the plough layer and the subsoil.

Derived Variates
To apply the method of auxiliary variables, we first defined the vector of variates x = (x1, x2, x3). For the pdfs of {theta}(h) and {theta}avail, we made w the first variate (x1), while for the pdf of K(h), x1 = k. In both cases, x2 = a and x3 = n. The pdf of the original variates, f(x1, x2, x3), is given by Eq. [3]. To find the pdf of {theta}(h), the Jacobian ({partial}{theta}/{partial}w) must be found by expressing Eq. [1] in terms of exp(w), exp(a), and exp[exp(n)], and then taking the derivative with respect to w. For the pdf of K(h), a similar procedure must be applied to Eq. [2]. Owing to the nature of Eq. [2] the resulting Jacobian {partial}K/{partial}k is simply equal to K. For the pdf of {theta}avail, the Jacobian equals {partial}{theta}avail/{partial}w. The expression of {theta}avail in terms of exp(w), exp(a), and exp[exp(n)] is

[11]

Inserting the joint pdf (Eq. [3]) of either w or k, and a and n, together with the required Jacobian, into Eq. [8] gives pdfs of {theta}(h), K(h), and {theta}avail of the type:

[12]
where q denotes the derived variate, p is either w or k, depending on the derived variate, and an overbar indicates the arithmetic mean of a variate. The function p(q) arises from the need to express w or k as a function of the variate for which the pdf is sought. The function can be found by first expressing the variate ({theta}(h), K(h), or {theta}avail) as a function of w or k using Eq. [1] or [2], and then inverting this relationship. In this study, all Jacobians could be expressed as exp[p(q)]A (with A a function defined below). For the pdf of {theta}(h), we have

[13a]

[13b]

[13c]

[13d]

[13e]

For the pdf of K(h), the required substitutions are

[14a]

[14b]

[14c]

[14d]

[14e]

Note that the Jacobian {i.e., exp[p(q)]A} indeed equals K(h), as was noted above. The pdf of {theta}avail is obtained as follows:

[15a]

[15b]

[15c]

[15d]

[15e]

The pdfs of the derived variates could be determined by numerically evaluating the double integral in Eq. [12]. We used a double Romberg scheme (Press et al., 1992, p. 134–136, 155–158). For decreasing h, the pdf of K(h) developed an extremely narrow peak and a very elongated right tail. Consequently, numerical evaluation was not always possible.

Figure 4 depicts those pdfs of K(h) that could be reliably calculated and fitted over the plotted range. For all pressure heads, the degree of variability is much larger in the undisturbed subsoil than in the plough layer. The logarithmic horizontal scale reflects the vastly increased skewness of the pdfs with decreasing h. The plotted range can only depict a section of the right tail for h ≤ –50 cm; the probability mass under dry conditions is concentrated around a mode that is orders of magnitude smaller than that of Ks.



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  		Fig. 4. Analytically derived pdfs of K(h) of the plough layer and the subsoil. Numbers indicate pressure heads in centimeters.

 
Figure 5 shows the pdfs of {theta}(h) of the plough layer and the subsoil. The pdf for h = –100 cm for the plough layer was remarkably peaked. The pdfs for h ≥ –100 cm were fairly symmetrical, while a more extensive right tail developed when the soil dried. A similar trend is visible for the subsoil, with the skewness reversing signs between h = –100 and –1000 cm. The left tail was completely lacking for h = –5000 cm. The transition in shape and location of the pdfs with decreasing h was more gradual for the subsoil. The subsoil was considerably drier than the plough layer for pressure heads ≤–1000 cm. For both layers, the pdf of {theta}s (given by the solid line for h = 0 cm) dominated the pdf for h = –10 cm.



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  		Fig. 5. Analytically derived pdfs of {theta}(h) of the plough layer and the subsoil. Numbers indicate pressure heads in centimeters.

 
The pdfs of the available water content for the plough layer and the subsoil (Fig. 6) were found to be smooth. The subsoil exhibited stronger variation, while the plough layer had a more asymmetrical pdf. Figures 4 through 6 clearly demonstrate the wide spectrum of shapes that the pdfs of the derived variates can assume. A lognormal pdf at saturation can lead to markedly different pdfs under drier conditions.



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  		Fig. 6. Analytically derived pdfs of {theta}avail of the plough layer and the subsoil.

 
Numerical Results
The mean, standard deviation, and coefficients of skewness ({gamma}1; Abramowitz and Stegun, 1964, Eq. [26.1.15]) and excess ({gamma}2; Abramowitz and Stegun, 1964, Eq. [26.1.16]) were determined for K(h), {theta}(h), and {theta}avail. These four statistics require the mean and the second through fourth central moments of a distribution. The four statistics were calculated directly from the vG parameters of the soil samples (Table 6), and from the Monte Carlo simulation (Table 7). The values of {gamma}1 and {gamma}2 indicate significant asymmetry and deviations from the normal pdf in all cases.


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  		Table 6. Descriptive statistics of features derived from the soil hydraulic parameters according to van Genuchten (1980) of Andelfingen soil samples. Values of {theta} and K (cm d–1) were calculated for the pressure heads indicated in parentheses. Available water ({theta}avail) is defined as {theta}(–333 cm) – {theta}(–15000 cm). The coefficients of skewness ({gamma}1) and excess ({gamma}2) describe the shape of the distribution [{gamma}1 < 0: left tail; {gamma}1 > 0: right tail; less ({gamma}2 < 0) or more peaked ({gamma}2 > 0) than a normal pdf].

 

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  		Table 7. Descriptive statistics of features derived from the Monte Carlo simulation based on the joint parameter pdf of the soil hydraulic parameters according to van Genuchten (1980). Values of {theta} and K (cm d–1) were calculated for the pressure heads indicated in parentheses. See also Table 6.

 
The Monte Carlo simulation reproduced the mean and standard deviation of K(h) in the plough layer reasonably well, but was less successful with the skewness and excess (compare the values of the mean, SD, {gamma}1, and {gamma}2 Table 6 [observed] and Table 7 [Monte Carlo]). The results for {theta}(h) in the plough layer were much better, with a minor overestimation of the standard deviation at h = –5000 cm. Note that {gamma}1 for {theta}(h) has the wrong sign for a dry plough layer. The analytically derived pdfs have the same problem: the right tails of Fig. 5 (plough layer) are not corroborated by the data from the soil samples.

The Monte Carlo simulation reproduced the mean of K(h) of the subsoil reasonably well, but performed somewhat more poorly for the standard deviations. At saturation, {gamma}1 was somewhat underestimated. Results for {gamma}2 were within an order of magnitude. The mean and standard deviation of {theta}(h) in the moist subsoil (h ≥ –100 cm) were reproduced well. For the drier subsoil the estimates deviated somewhat more, but in a consistent manner. The Monte Carlo simulation produced values of {gamma}2 that were consistently too low, and frequently had an incorrect sign. In dry soils, the estimates of both {gamma}1 and {gamma}2 were poor, and the sign was frequently incorrect also.

In general, the differences of the distribution of {theta}avail between the plough layer and the subsoil that are visible in Table 6 were reproduced rather well by the Monte Carlo simulation, especially for the lower-order moments (mean and standard deviation).


   SUMMARY AND CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSIONS
REFERENCES
 
We introduced a procedure that uses curvilinear regression analysis to identify possible relations between the vG parameters. We then estimated pdfs of the transformed parameters and the correlations between them. The correlations were too low in most cases to allow the use of Ks (or any other parameter) as a predictor of another (transformed) parameter. Still, our results show that if at least one parameter has a well-defined pdf, selecting a transformation of another parameter on the basis of regression analysis between both parameters can help one select the proper type of pdf of the parameter under consideration. Choosing pdfs in this way highlights the correlation between parameters, which allows a reliable assessment of the interdependence among parameters. In contrast, independent fitting of pdfs may lead to less suitable parameter pdfs that could mask possible correlations among parameters by R2 values that are too low.

The analysis of the variation and correlation of the soil hydraulic parameters benefited enormously from the fact that the joint distribution of their transformations was multivariate normal. This normal joint pdf was a direct consequence of the fact that the pdf of lnKs was normal. Since Ks has been found to be lognormally distributed in many soils, the procedures used in this paper that rely on the multivariate normality of the joint pdf may well be generally applicable to soils around the world. Also, since Ks is usually the most variable parameter, the procedure introduced here to identify suitable parameter transformations is probably applicable to any other parameterization that includes Ks.

Another key advantage of multivariate normal distributions of the transformed vG parameters is the relative ease by which pdfs of derived variates (functions of the vG parameters) can be determined, either analytically or through Monte Carlo simulation. Again, the fact that the pdf of Ks is lognormal for most soils makes these techniques widely applicable. The analytical pdfs of the derived variates were mathematically vigorous and clarified the relationships between the derived variates and the soil hydraulic properties. Evaluating these pdfs proved difficult because of the need for a numerical multiple integration that was computationally demanding and compromised accurate evaluation of the pdfs of dry soils. The Monte Carlo simulation, while lacking a clear mathematical link between the derived variates and the soil hydraulic properties, was easily implemented, computationally efficient, and robust.

Plots of the derived variates and the skewness and excess calculated from the data and the Monte Carlo simulation all showed markedly nonsymmetric pdfs. Simplifying a statistical analysis by only considering the first two moments may therefore not always provide useful information.


   ACKNOWLEDGMENTS
 
The research of G.H. de Rooij has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. We thank Prof. Alfred Stein of Wageningen University for his suggestions and remarks, which helped to improve the paper.


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 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSIONS
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